A FULLY COUPLED NAVIER-STOKES SOLVER FOR FLUID …pressure-based algorithm for the solution of fluid flow at all speeds. The new algorithm is an extension into compressible flows

Copyright © Taylor & Francis Group, LLCISSN: 1040-7790 print/1521-0626 onlineDOI: 10.1080/10407790.2013.869102

A FULLY COUPLED NAVIER-STOKES SOLVER FORFLUID FLOW AT ALL SPEEDS

M. Darwish and F. MoukalledDepartment of Mechanical Engineering, American University of Beirut, Beirut,Lebanon

This article deals with the formulation and testing of a newly developed, fully coupled,pressure-based algorithm for the solution of fluid flow at all speeds. The new algorithmis an extension into compressible flows of a fully coupled algorithm developed bythe authors for laminar incompressible flows. The implicit velocity–pressure–densitycoupling is resolved by deriving a pressure equation following a procedure similar toa segregated SIMPLE algorithm using the Rhie-Chow interpolation technique. Thecoefficients of the momentum and continuity equations are assembled into one matrixand solved simultaneously, with their convergence accelerated via an algebraic multigridmethod. The performance of the coupled solver is assessed by solving a number of two-dimensional problems in the subsonic, transsonic, supersonic, and hypersonic regimesover several grid systems of increasing sizes. For a desired level of convergence, resultsfor each problem are presented in the form of convergence history plots, tabulated valuesof the maximum number of required iterations, the total CPU time, and the CPU timeper control volume.

INTRODUCTION

Despite its wide adoption and success in solving a broad range of flowproblems encompassing incompressible [1, 2] and compressible fluid flow at allspeeds [3–8], single [9, 10] and multiphase flows [11, 12], laminar [13] and turbulentflows [14–16], free-surface flows [17, 18], and particle-laden flows [19–21], to citea few, the segregated pressure-based approach [22, 23], built on the SIMPLEfamily of algorithms [24–26], continues to suffer from a breakdown in convergencerate for large-scale problems. The reason for incompressible flow problems failingto scale linearly with grid size is the weak pressure–velocity coupling that thesealgorithms use when discretizing the highly coupled Navier-Stokes equations, whichin compressible flows extend to the pressure–velocity–density coupling. One wayto address this weakness is through the use of the full algebraic storage (FAS)multigrid method [27–29]. Another approach is to ensure that the numerical

Received 24 September 2013; Accepted 7 November 2013.The financial support provided by the University Research Board of the American University

of Beirut is gratefully acknowledged.Address correspondence to F. Moukalled, Department of Mechanical Engineering, American

University of Beirut, P.O. Box 11-0236, Riad El Solh, Beirut, 1107 2020, Lebanon. E-mail:[email protected]

410

Numerical Heat Transfer, Part B, 45: 410–444, 2014

NAVIER-STOKES SOLVER FOR FLUID FLOW AT ALL SPEEDS 411

NOMENCLATURE

auuC � auv

F � � � � coefficients in the discretized equations � dynamic viscosityb�P source term in the discretized equations � fluid density

B source term in the momentum equation � deviatoric stress tensorC main grid point SubscriptsdCF vector joining the grid points C and F C refers to main grid pointD operator used in the pressure equation f refers to control-volume faceE component of the surface vector in the F refers to the F grid point

direction of dCF nb refers to values at the faces obtainedF refers to neighbor of the C grid point by interpolation between C and itsg geometric interpolation factor neighborsi� j unit vectors in the x and y directions, NB refers to the neighbors of the C

respectively grid point

mf mass flow rate at control-volume face f Superscriptsp pressure p refers to pressureP main grid point u refers to the u-velocity componentQ general source term v refers to the v-velocity componentRMS root-mean-square residuals � refers to value at the previous time stepS surface vector ∗ refers to value at the previous iterationt time ∗∗ refers to value at the current iterationT component of the surface — refers to an interpolated value

vector to Su� � velocity components in x and y

directions, respectivelyv velocity vector

discretization reflects the strong coupling that exists in the Navier-Stokes equations.In this case all equations are solved simultaneously in a coupled fashion as opposedto a segregated approach. As described in [30], such an approach is actually usedin density-based methods [31, 32], where the Navier-Stokes and energy equationsare solved simultaneously as one system of equations. It is worth mentioning thatthe Computational fluid dynamics (CFD) group at Imperial College, originator ofthe SIMPLE algorithm, had initially developed a coupled pressure-based solver[33] and not a segregated solver. However, their coupled algorithm, known asSIVA, was overshadowed by the SIMPLE algorithm, which combined low memoryrequirement with coding simplicity, two substantial advantages given the stateof computer technology at that time. Recent advances in computer architecturerenewed interest in coupled algorithms and led to the development of a numberof new approaches [34–37]. A coupled algorithm for incompressible flows has beensuccessfully implemented by the authors [38, 39], showing near-linear scalability forgrids in the range 10,000 to 300,000 elements. The current work extends the coupledalgorithm to compressible flow situations. This extension requires mass conservationto be expressed not only by a change in pressure, but also by a change in density.Therefore the algorithm must express this relationship in its numerics to be ableto resolve supersonic, transsonic, or even subsonic flows at a Mach number greaterthan about 0.3.

The compressible pressure-based coupled algorithm is presented next by firstdetailing the finite-volume discretization of its coupled equations. Then a set of

412 M. DARWISH AND F. MOUKALLED

problems is presented that illustrate the convergence and computational cost ofthe coupled solver. The performance is assessed by solving a number of two-dimensional problems over several grid systems of increasing size and noting thesolver performance to convergence in terms of number of iterations, CPU time, andcomputational cost per control volume.

DISCRETIZATION OF INTERNAL ELEMENTS

In the coupled approach the momentum and mass conservation equations aresolved simultaneously. The mass conservation equation is not used in its originalform; rather, it is transformed into a pressure equation that is coupled to the velocityfield of the momentum conservation equation. Thus the velocity and pressurefields are calculated simultaneously based on guessed or estimated velocity andpressure fields to obtain pressure and velocity fields that satisfy both the momentumconservation and mass conservation. This technique guarantees momentum andmass conservation at any iteration. The discretization process proceeds as follows:The solution domain is first discretized by subdividing it into a number of controlvolumes with each one associated with a main grid point C (Figure 1) placed at itscentroid. Then the conservation equations are discretized using a two-step procedureas described next for the various equations.

Momentum Equation

The vector form of the momentum equation can be written as

� �v�t

+ � · �vv = −�p+ � · �+ B (1)

Figure 1. A schematic of a control volume C, its neighbor control volume F , and related geometricquantities.


The discretization process starts by first integrating Eq. (1) over the control volumeshown in Figure 1, to yield

∫∫

� �v�t

d +∫∫

� · �vv d = −∫∫

�pd +∫∫

� · � d +∫∫

B d (2)

where is the volume of the control volume whose surface is denoted by � . Thevolume integrals are then transformed into surface integral using the divergencetheorem. This results in

∫∫

� �v�t

d +∮�

�vv · dS = −∮�

p dS+∮�

� · dS+∫∫

Bd (3)

Adopting an implicit Euler integration scheme for the transient term and evaluatingsurface and volume integrals using a second-order interpolation profile (trapezoidalrule), Eq. (3) is transformed to

�v− �v�

�t C + ∑

f=nb�C

�vv − �f · Sf +∑

f=nb�C

pfSf = BC C (4)

All terms appearing in Eq. (4) are evaluated at time t except for the term �v�,which is evaluated at time t − �t. Finally, the equation is transformed into analgebraic equation by expressing the variation in the dependent variable and itsderivatives in terms of the grid-point values. The resulting equation links thevalue of the dependent variable at the control-volume center to the neighboringdependent-variable values and is given by

avvC vC + ∑

F=NB�C

avvF vF + a

vpC pC + ∑

F=NB�C

avpF pF = bvC (5)

where

avvC = P

�t+ ∑

f=nb�C

(��Ef��dCF � +

∥∥mf � 0∥∥) avv

F = −��Ef��dCF � −

∥∥−mf � 0∥∥

avpC = gCSf a

vpF = gFSf bvC = BC C + �v�

�t C − ∑

f=nb�C

(�� · Tf +

∥∥mf � 0∥∥) (6)

The vectors dCF � Ef � and Tf are displayed in Figure 1. When the interest is in asteady-state solution, the transient contribution to the bvC term is modified into afalse transient term and instead of the old value the previous value of the computedfield is used as

bvC = BC C + �v∗

�t C − ∑

f=nb�C

(�� · Tf +

∥∥mf � 0∥∥) (7)


Continuity Equation

The semidiscrete form of the continuity equation can be written as

− �

�t C + ∑

f=nb�C

mf = 0 (8)

In this equation the pressure and density fields are related by the ideal gas relation as

=(

1RT

)p = Cp (9)

At any point the density, velocity, and pressure fields can be defined as ∗� v∗, andp∗, with these fields not completely satisfying the momentum and mass conservationequations. The aim is to derive updated fields ∗∗� v∗∗, and p∗∗ that better satisfythese conservation equations. Using the updated fields, the continuity equation isgiven by

∗∗C − �

C

�t C + ∑

f=nb�C

∗∗f v∗∗f · Sf = 0 (10)

Applying a Newtonian linearization, the change in fvf with iterations (n can bewritten as

d(fvf

)∗dn

=(fvf

)∗∗ − (fvf)∗

�n+ 1− n= ∗

f

dv∗fdn

+ v∗fd∗

f

dn∗∗f v∗∗f − ∗

fv∗f

= ∗f

(v∗∗f − v∗f

)+ v∗f(∗∗f − ∗

f

)⇒ ∗∗f v∗∗f = ∗

fv∗∗f + ∗∗

f v∗f − ∗fv

∗f (11)

Substituting the value of ∗∗f v∗∗f from Eq. (11) into Eq. (10), the continuity equation

becomes

∗∗C − �

C

�t C + ∑

f=nb�C

∗

fv∗∗f · Sf +

compressible︷︸︸︷∗∗f v∗f · Sf − ∗

fv∗f · Sf

= 0 (12)

where the terms labeled “compressible” in Eq. (12), and others presented later, areadditional terms arising due to compressibility effects. Using the ideal gas relation,Eq. (12) is changed to

C∗p

∗∗C − �

C

�t C + ∑

f=nb�C

∗

fv∗∗f · Sf +

compressible︷︸︸︷C∗

m∗f

∗f

p∗∗f − m∗

f

= 0 (13)

The discretization of the continuity equation needs further treatment in order toderive the pressure equation. This is because, unlike the momentum equation, whichis a primitive equation for velocity, continuity acts as a constraint on the pressure


field. For a collocated grid system, coupling of the pressure and velocity fields isensured by computing the velocity at the control-volume faces using the Rhie-Chowinterpolation [3], according to which the interface velocity is given by

vf = vf −Df

(�pf − �pf

)(14)

where the averaged quantityfis computed using linear interpolation as

vf = gCvC + gFvF �pf = gC�pC + gF�pF Df = gCDC + gFDF gC + gF = 1 (15)

Substituting Eq. (14) into Eq. (13), the continuity equation after manipulationbecomes

compressible︷︸︸︷C∗

C

�tp∗∗C + ∑

f=nb�C

(C∗

∗f

m∗fp

∗∗f

)+ ∑

f=nb�C

[∗fv

∗∗f − ∗

fDf�p∗∗f

] · Sf

=

compressible︷︸︸︷ C

�t�C + ∑

f=nb�C

m∗f −

∑f=nb�C

(∗fDf�p

∗f · Sf

)(16)

Combining this pressure equation with the momentum equation [Eq. (5)] yields thefollowing system of equation with v∗∗ and p∗∗ as variables:

[avvC a

vpC

apvC a

ppC

] [v∗∗Cp∗∗C

]+ ∑

F=NB�C

[avvF a

vpF

apvF a

ppF

] [v∗∗Fp∗∗F

]=[bvCbpC

](17)

For a two-dimensional flow problem this is written as

auu

C auvC a

upC

avuC avv

C avpC

apuC a

pvC a

ppC

u∗∗

C

v∗∗Cp∗∗C

+ ∑

F=NB�C

auu

F auvF a

upF

avuF avv

F avpF

apuF a

pvF a

ppF

u∗∗

F

v∗∗Fp∗∗F

=

buCbvCbpC

(18)

where the coefficients are given as follows.Coefficients for the u-velocity equation:

auuC = ∗

C C

�t+ ∑

f=nb�C

(�f

∥∥Ef

∥∥�dCF�

+ ∥∥m∗f � 0

∥∥) auuF = −�f

∥∥Ef

∥∥�dCF�

− ∥∥−m∗f � 0

∥∥auvC = 0 auv

F = 0 aupC = ∑

f=nb�C

gfCSf · i aupF = gfFSf · i

buC = ∗Cu

�C

�t C + BC · i C + ∑

f=nb�C

(�f�u

∗f · Tf

)+ ∑f=nb�C

[13�f

(� · v∗f

)i · Sf

](19)


Coefficients for the v-velocity equation:

avvC = ∗

C C

�t+ ∑

f=nb�C

(�f

∥∥Ef

∥∥�dCF�

+ ∥∥m∗f � 0

∥∥) avvF = −�f

∥∥Ef

∥∥�dCF�

− ∥∥−m∗f � 0

∥∥avuC = 0 avu

F = 0 avpC = ∑

f=nb�C

gfCSf · j avpF = gfF

(Sf · j

)bvC = ∗

Cv�f

�t C + �BC · j C + ∑

f=nb�C

(�f�v

∗f · Tf

)+∑f=nb�C

[13�f

(� · v∗f

)j · Sf

](20)

Coefficients for the pressure equation:

appC = C

RT ∗C�t

+ ∑f=nb�C

(Df

∥∥Ef

∥∥�dCF�

+∥∥∥∥ m∗

f

∗f

C� 0

∥∥∥∥)

appF = −Df

∥∥Ef

∥∥�dCF�

−∥∥∥∥− m∗

f

∗f

C� 0

∥∥∥∥apuC = ∑

f=nb�C

gfCSf · i apuF = gfF

(Sf · i

)apvC = ∑

f=nb�C

gfCSf · j apvF = gfF

(Sf · j

)bpC = − ∑

f=nb�C

m∗f +

∑f=nb�C

(Df�p

∗f · Tf

)(21)

DISCRETIZATION OF BOUNDARY ELEMENTS

A boundary element has at least a boundary face where boundary conditionsare applied. The treatment of these boundary conditions is critical to the accuracyand robustness of any numerical scheme. For coupled algorithms it greatly affectsthe convergence history, since at these boundaries additional coupling needs to betaken into account. For example, the auv and avu coefficients are nonzero at wallboundaries; also, the apu and apv coefficients are greatly affected by the type ofboundary condition at inlets and outlets. The general discretization of the continuityand momentum equations along a boundary element is presented next. The fulldiscretization of the different types of boundary conditions will be the subject of afuture article.

Momentum Equation

The semidiscretized form of the momentum equation can be expressed as

�vC − �v�C�t

P︸︷︷︸element discretization

+ ∑f=nb�C

(mfvf

)︸︷︷︸face discretization

= − ∑f=nb�C

(PfSf


+ ∑f=nb�C

(�f · Sf


+ B︸︷︷︸element discretization

(22)

where the discretization type of the various terms is explicitly stated. Terms thatare evaluated at the control-volume faces should be modified along a boundary facein accordance with the type of boundary condition used. Therefore, for boundary


elements, the terms evaluated at the control-volume faces are written as

∑f=nb�C

(mfvf


= ∑f=interior nb�C

(mfvf

)+ mbvb︸︷︷︸boundary term∑

f=nb�C

(�f · Sf



(�f · Sf

)+ �b · Sb︸︷︷︸boundary term


(�f · Sf

)+ Fbboundary term∑

f=nb�C

(PfSf



(PfSf

)+ PbSbboundary term

(23)

where subscript b refers to value at the boundary.

Continuity Equation

The semidiscretized form of the continuity equation can be stated as

C − �C

�t C︸︷︷︸

element discretization

+ ∑f=nb�C

mf︸︷︷︸face discretization

= 0 (24)

Similar to the momentum equation, for boundary elements the term evaluated atthe control-volume face is written as

∑f=nb�C

mf︸︷︷︸face discretization


(mf

)+ mb︸︷︷︸boundary term


(fvf · Sf

)

+ bvb · Sb︸︷︷︸boundary term

(25)

Note that for the case of compressible flow a Newtonian linearization is applied toyield

∑f=nb�C

(∗∗f v∗∗f · Sf

)︸︷︷︸

face discretization


(∗fv

∗∗f · Sf

)+ ∑f=interior nb�C

(∗∗f v∗f · Sf

)

− ∑f=interior nb�C

(∗fv

∗f · Sf

)+ ∗

bv∗∗b · Sb + ∗∗

b v∗b · Sb − ∗bv

∗b · Sb︸︷︷︸

boundary term

(26)


with the implementation taking the form∑f=nb�C

(∗∗f v∗∗f · Sf

)︸︷︷︸

face discretization


(∗fv

∗∗f · Sf

)

+

compressible︷︸︸︷∑f=interior nb�C

(m∗

f

∗f

)∗∗f − ∑

f=interior nb�C

m∗f + ∗

bv∗∗b · Sb +

compressible︷︸︸︷(m∗

b

∗b

)∗∗b − m∗

b︸︷︷︸boundary term

(27)

THE RHIE-CHOW INTERPOLATION

As mentioned earlier, the interface velocity is computed using the Rhie-Chowinterpolation technique [3] such that the mass flow rate is calculated as

m∗∗f = ∗

f · Sf +


f

∗f

)∗∗f − m∗

f = ∗f

(v∗∗f −D

(� p∗∗

f − � p∗∗f

) ) · Sf

+


f

∗f

)∗∗f − m∗

f (28)

At boundary faces the treatment of the velocity depends on whether the mass flowrate or the pressure is the specified variable. Generally three types of boundaryconditions are distinguished. The first type can be designated by “specified massflow rate” (e.g., walls, velocity specified at inlets for incompressible flow). For thiscategory, mb has a specified value and no modification to the pressure equation isneeded. This yields a Von Newman–like boundary condition for the pressure, withthe mass flux �mb being specified. The pressure, however, has to be computed atthe boundary from the interior field.

The second type is “pressure specified,” where pb has a specified value. For thisboundary condition type, pb is written in terms of the velocity vector and pressuregradient of the nearest element, yielding a Dirichlet-like condition that should beenforced on the pressure equation.

In the third type, an implicit relation exists between the pressure and the massflow rate, as in a specified total pressure boundary condition. In this case, an explicitequation is extracted from the implicit relation and substituted into the pressureequation, yielding a hybrid boundary condition.

The type of pressure boundary condition also affects the treatment of thepressure gradient term in the momentum equations. One more detail of interestis the expression for the Rhie-Chow interpolation at the boundary faces. Forboundary faces the averaging in the Rhie-Chow interpolation is written in terms ofthe boundary cell only as

b=

C(29)


Thus

v∗b = v∗C �p�nb = �p

�nC Db = DC � � � (30)

and the Rhie-Chow interpolation at the boundary becomes

v∗∗bboundary face

= v∗∗C −DC ��p∗∗b − �p∗∗

C ︸︷︷︸boundaryRhie−Chow

(31)

with the mass flow rate computed as

m∗∗b = ∗

b

(v∗∗c −Dc �� p∗∗

b − � p∗c)· Sb +


b

∗b

)∗∗b − m∗

b

= ∗bv

∗∗C · Sb −DC

�Eb��dCb�

�p∗∗b − p∗∗

C − �DC�p∗∗b · Tb −DC�p

∗∗C · Sb

+


b

∗b

)∗∗b − m∗

b (32)

RESULTS AND DISCUSSION

The validity and performance of the coupled algorithm is assessed in thissection by presenting solutions to the following steady inviscid two-dimensionaltest cases: (1) flow over a NACA0012 airfoil; (2) flow over a circular arc bump;(3) supersonic flow over an obstacle; (4) supersonic flow over a circular cylinder;(5) hypersonic flow over a wedge; (6) and flow in a converging-diverging nozzle.Each problem is solved, starting from the same initial guess, over several gridsystems of increasing density using triangular and quadrilateral elements. For allproblems, computations were terminated when the normalized root-mean-square(RMS) residuals, defined as

RMS = 1N

√√√√ N∑i=1

(a�C�C+

∑F=NB�C

a�F �F−b

�C

a�C�scale

)2

N = number of elements

�scale = max(�C�max − �C�min� �C�max

)�C�max = N

maxi=1

��C �C�min =N

mini=1

��C

(33)

over the domain and for all dependent variables fell below 10−5. All computationswere performed on a MacPro computer with a 2.8-GHz Intel Xeon processor.

Problem 1: Flow over a NACA0012 Airfoil

The first test case considered is for the steady flow around a NACA 0012airfoil, which is a standard test case used by several researchers to validate CFDcodes [28, 40–44]. For every grid system considered, two flow conditions aresimulated. In the first, the flow approaching the airfoil, shown schematically in


Figure 2a, is at a Mach number of 0.63 and an angle of attack �= 2�. Under theseconditions the flow over the airfoil is fully subsonic. In the second configuration,the free-stream Mach number is set at 0.85 and � at 1� resulting in a transsonicflow with shock waves forming on the upper and lower sides of the airfoil. Foreach flow condition, the problem is solved using quadrilateral elements over threegrid systems with sizes of 34,000, 136,000, and 406,000 control volumes. Solutionsare also generated using triangular elements over three grid networks with sizesof 43,000, 120,000, and 370,000 control volumes. An illustrated grid generated isdepicted in Figure 2b.

For subsonic flow conditions isobars are presented in Figure 2c, whileconvergence history plots of the continuity, momentum, and energy equations forthe various grid systems and element types are displayed in Figure 3. The reductionof residuals with number of iterations for quadrilateral elements is depicted inFigures 3a–3c, while those for triangular elements are presented in Figures 3d–3f .As shown, the convergence paths for all cases are similar.

A summary of the number of iterations, CPU time, and CPU time per controlvolume are presented for all grid sizes in Table 1a. As depicted, the numberof iterations varies between 24, for a grid size of 34,000 quadrilateral controlvolumes, and 79, for a grid size of 370,000 triangular control volumes. The CPUtime increases from 233 s for the case of 34,000 quadrilateral control volumes to16,289 s for the case of 370,000 triangular control volumes. A more indicativeperformance parameter is the CPU per control volume, which for a quadrilateralelement increases from 6�85× 10−3 s to 31�46× 10−3s when the grid size increasesfrom 34,000 to 406,000 control volumes. This represents around a 359.27% increasein the solution cost per control volume for a 1,094.12% increase in the mesh size.For a triangular element, the CPU per control volume increases from 16�26× 10−3sto 44�02× 10−3s as the grid size increases from 43,000 to 370,000 control volumes.

Results for the transsonic flow case are presented in Figures 4 and 5. Figure4a depicts isobars around the airfoil, with the shocks developing on the upper andlower surfaces of the airfoil clearly seen. Moreover Figure 4b presents a comparisonof the variation of the pressure coefficient on the upper and lower sides of theairfoil between values generated using the coupled solver and similar ones reportedby Favini [40]. As shown, Cp results are on top of each other, indicating a correctimplementation of the newly developed coupled method. Again, the convergencehistory plots over the various grid systems and element types are displayed inFigure 5. The reduction of residuals with number of iterations for quadrilateralelements is depicted in Figures 5a–5c, while those for triangular elements arepresented in Figures 5d–5f . As shown, the convergence paths for all cases aresimilar.

A summary of the number of iterations, the CPU time, and CPU time percontrol volume are presented for all grid sizes in Table 1b. As depicted, the numberof iterations in this case is greater than the number of iterations required in thesubsonic case due to the more complex flow to be resolved. The number of iterationsvaries between 117, for a grid with size of 34,000 quadrilateral control volumes,and 733, for a grid with size of 370,000 triangular control volumes. The CPUtime increases from 1,288 s for the case of 34,000 quadrilateral control volumes to150,690 s for the case of 370,000 triangular control volumes. At the same time, the


Figure 2. (a) The physical situation for the flow around a NACA 0012 airfoil; (b) an illustrativegrid in the region close to the airfoil; and (c) isobars for subsonic flow over a NACA 0012 airfoil(M� = 0�63� � = 2�.


Figure 3. Convergence history for subsonic flow over a NACA0012 airfoil (M� = 0�63� � = 2�.

CPU per control volume for quadrilateral elements increases from 37�88× 10−3 s to254�46× 10−3 s when the grid size increases from 34,000 to 406,000 control volumes,while for triangular elements it increases from 9�14× 10−3 s to 407�27× 10−3 s asthe grid size increases from 43,000 to 370,000 control volumes. Nevertheless, it waspossible to obtain solutions with all grid systems used.

Problem 2: Flow over a Circular Arc Bump

The physical situation shown schematically in Figure 6a represents a channelof width equal to the length of the circular arc bump and of total length equal tothree lengths of the bump [10, 40, 45]. Results are presented for flow in the subsonic,transonic, and supersonic regimes. For subsonic and transsonic calculations, thethickness-to-chord ratio is 10%, and for supersonic flow calculations it is 4%. Whenavailable, predictions are compared with published data.

Case 1: Subsonic flow over a circular arc bump. With an inlet Machnumber of 0.5, the inviscid flow in the channel is fully subsonic and almostsymmetric across the middle of the bump. At the inlet, the flow is assumed to haveuniform properties, and all variables, except pressure, are specified. At the outletsection, the pressure is prescribed and all other variables are extrapolated from theinterior of the domain. The flow tangency condition is applied at the walls.

Figure 6b displays Mach contours over the domain, while Figure 6c presents acomparison between predicted Mach number values over the lower and upper wallsof the domain with similar ones reported by Favini [40]. As shown, results are inexcellent agreement.


The total computational cost, the cost per control volume, and the number ofiterations required to obtain converged solutions over the five grid sizes used arepresented in Table 1c. When using quadrilateral control volumes, the number requiredto obtain a converged solution for all grid sizes is between 20 and 22 iterationsexcept for a grid with size of 50,000 Control Volumes (CV) where 32 iterations arerequired. For triangular elements, the number of iterations required with all grid sizesis 11 (except for the 10,000 CV grid where 13 iterations are needed). The CPU timeper control volume varies between 6�00× 10−3s and 11�18× 10−3s for quadrilateralelements and between 2.0×10−3s and 4.63×10−3s for triangular elements.

Table 1. (a) Iterations and CPU time for subsonic flow over a NACA0012 airfoil (M� = 0�63� �= 2�,(b) Iterations and CPU time for transsonic flow over a NACA 0012 airfoil (M� = 0�85� �= 1�, (c)Iterations and CPU time for subsonic flow over a circular arc bump (Min = 0�5), (d) Iterations andCPU time for flow over a circular arc bump (transconic, Min = 0�675), (e) Iterations and CPU timefor flow over a circular arc bump (supersonic, Min = 1�4), (f ) Iterations and CPU time for flow overa circular arc bump (supersonic, Min = 1�65)

Quadrilateral elements Triangular elements

Grid Size No. of #Iter. CPU�s CPU/CV Grid Size No. of #Iter. CPU�s CPU/CV

(a)34,000 24 233 6�85× 10−3 43,000 30 699 16�26× 10−3

136,000 43 3� 718 27�34× 10−3 120,000 32 2� 185 18�21× 10−3

406,000 48 12� 775 31�46× 10−3 370,000 79 16� 289 44�02× 10−3

(b)34,000 117 1� 288 37�88× 10−3 43,000 35 393 9�14× 10−3

136,000 272 23� 470 172�57× 10−3 120,000 189 12� 884 107�37× 10−3

406,000 392 104� 116 256�44× 10−3 370,000 733 150� 690 407�27× 10−3

(c)1,000 20 6 6�00× 10−3 1,000 11 2 2�00× 10−3

10,000 22 68 6�80× 10−3 10,000 13 39 3�9× 10−3

50,000 32 559 11�18× 10−3 50,000 11 212 4�24× 10−3

100,000 23 884 8�84× 10−3 100,000 11 434 4�34× 10−3

300,000 21 2� 830 9�43× 10−3 300,000 11 1� 390 4�63× 10−3

(d)1,000 18 6 6�00× 10−3 1,000 16 3 3�00× 10−3

10,000 39 106 10�60× 10−3 10,000 19 57 5�7× 10−3

50,000 90 1� 340 26�80× 10−3 50,000 22 409 8�18× 10−3

100,000 135 4� 096 40�96× 10−3 100,000 100 2� 580 25�80× 10−3

300,000 97 7� 515 25�05× 10−3 323,000 85 6� 958 21�54× 10−3

(e)10,000 24 67 6�70× 10−3 10,000 25 209 20�90× 10−3

66,000 31 595 9�02× 10−3 66,000 35 1� 246 18�88× 10−3

333,000 57 9� 870 29�64× 10−3 333,000 62 11� 538 34�65× 10−3

(f )10,000 23 63 6�30× 10−3 10,000 20 167 16�70× 10−3

66,000 27 520 7�88× 10−3 66,000 24 851 12�89× 10−3

333,000 36 6� 221 18�68× 10−3 333,000 36 6� 631 19�91× 10−3


Figure 4. (a) Isobars and (b) comparison of predicted pressure coefficient values along the upper andlower surfaces of the airfoil with published data [40] for transsonic flow over a NACA 0012 airfoil(M� = 0�85� � = 1�.

The reduction of residuals with number of iterations for quadrilateral elementsis depicted in Figures 7a–7c, while reduction for triangular elements is revealed infigures 7d–7f . As shown, for a given type of grid the convergence paths for all casesare similar.

Case 2: Transsonic flow over a circular arc bump. With the exceptionof the inlet Mach number being set at 0.675, the grid distribution and the


Figure 5. Convergence history for transsonic flow over a NACA0012 airfoil (M� = 0�85� � = 1�.

implementation of boundary conditions are identical to those described for subsonicflow. Results are displayed in terms of lines of constant Mach number over thedomain Figure 8a and Mach profiles along the walls Figure 8b. Predicted profilesare compared with similar ones reported in [40] and shown to be in excellentagreement. In addition, the transonic nature of the flow is demonstrated by theformation of the shock wave, which is clearly shown in Figures 8a and 8b.

The performance of the coupled solver as the number of grid points increasescan be inferred from the values presented in Table 1d. With quadrilateral controlvolumes, the number of iterations needed for convergence varies between 18 and135. For triangular elements, the number varies between 16 and 100. The CPU timeper control volume varies between 6�00× 10−3s and 40�96× 10−3s for quadrilateralelements and between 3�0× 10−3s and 25�8× 10−3s for triangular elements. Thedecrease in residuals with iteration is presented in Figures 9a–9c for grid systemsusing quadrilateral elements and in Figures 9d–9f for grid networks employingtriangular elements. Plots indicate that the convergence paths are similar for a giventype of elements.

Case 3: Supersonic flow over a circular arc bump. Computations areperformed for two values of the inlet Mach number, 1.4 and 1.65. For these valuesand for the geometry used, the flow is also supersonic at the outlet. Thus, allvariables at the inlet are prescribed, and at outlet all variables are extrapolated.Mach contours over the domain are presented in Figure 10a for Min = 1�65, and inFigure 10b for Min = 1�4. These contours are in excellent agreement with similar onesreported in the literature. For further confirmation, the predicted Mach values alongthe lower and upper walls are compared in Figure 10c with similar ones reported


Figure 6. (a) Physical situation for the flow over a circular arc bump; (b) Mach contours for subsonicflow over a circular arc bump (Min = 0�5); and (c) comparison of predicted Mach number values alongthe upper and lower walls with published data [40] (Min = 0�5).


Figure 7. Convergence history for subsonic flow over a circular arc bump (Min = 0�5).

in [40]. As shown, profiles are on top of each other, indicating once more the correctimplementation of the coupled algorithm.

Results reported in Table 1e indicate that for Min = 1�4 the number ofiterations required for convergence varies between 24 and 57 as the number ofquadrilateral elements increases from 10,000 to 333,000. The computational timeper control volume varies between 6�70× 10−3s and 29�64× 10−3s. For triangularelements, the number of iterations varies between 25 and 62, while the CPU timeper control volume changes between 20�90× 10−3s and 34�65× 10−3s as the numberof grid points increases from 10,000 to 333,000. The convergence history is depictedin Figures 11a–11c for grid systems using quadrilateral elements and in Figures11d–11f for grid networks employing triangular elements. Plots indicate that theconvergence paths are to some extent similar for a given type of elements.

At the higher inlet Mach number (Min = 1�65), convergence data over thevarious grid systems used are presented in Table 1f . As depicted, the number ofiterations varies between 23, for a grid with size of 10,000 quadrilateral controlvolumes, and 36, for a grid with size of 333,000 quadrilateral/triangular controlvolumes. The CPU time increases from 63 s for the case of 10,000 quadrilateralcontrol volumes to 6,631 s for the case of 333,000 triangular control volumes.Moreover, the CPU per control volume for a quadrilateral element increases from6�30× 10−3 s to 18�68× 10−3s when the grid size increases from 10,000 to 333,000control volumes. This represents a 196.51% increase in the solution cost per controlvolume for 3,230 % increase in the mesh size. For a triangular element, the CPUper control volume increases from 16�70× 10−3 s to 19�91× 10−3 s as the grid sizeincreases from 10,000 to 333,000 control volumes. The increase in computationalcost in this case is 19.22% for 3230 % increase in mesh size. The convergence


Figure 8. (a) Mach contours for transsonic flow over a circular arc bump (Min = 0�675); and (b)comparison of predicted Mach number values along the upper and lower walls with publisheddata [40] (Min = 0�675).

history plots are displayed in Figures 12a–12c for grid networks using quadrilateralelements and in Figures 12d–12f for grid systems using triangular elements. Plotsindicate that the convergence paths are mostly similar.

Problem 3: Supersonic Flow over an Obstacle

The physical situation is depicted in Figure 13a and represents a channel ofheight H and width W with a baffle of height h=H/9 mounted to its base and


Figure 9. Convergence history for transsonic flow over a circular arc bump (Min = 0�675).

located at its middle. Solutions are generated over a number of grid systems for thecase when air enters the domain at a Mach number of 2 (Min = 2). Predicted isobarsare displayed in Figure 13b.

The convergence history plots are displayed in Figures 14a–14c for gridnetworks using quadrilateral elements and in Figures 14d–14f for grid systems usingtriangular elements. Plots indicate that for a given type of element, the convergencepaths are similar for the different grid sizes. Moreover, plots in Figures 14a–14cshow smooth convergence, whereas plots in Figures 14d–14f reveal a very roughconvergence in the initial stage of the iterative process and then the convergenceproceeds smoothly. To be noticed also is the greater number of iterations requiredfor convergence to be reached, in comparison with the number required in theprevious problems. This increase in computational cost is related to the complexstructure of the flow.

Convergence data for the various grid systems used are presented in Table 2a.As depicted, the number of iterations varies between 180, for a grid with size of10,092 quadrilateral elements, and 861, for a grid with size of 314,094 quadrilateralcells. Simultaneously the CPU time increases from 748 s for the case of 10,092quadrilateral control volumes to 203,039 s for the case of 314,094 quadrilateralcontrol volumes. Moreover, the CPU per control volume for a quadrilateral elementincreases from 74�12× 10−3 s to 646�43× 10−3s when the grid size increases from10,092 to 314,094 control volumes. This represents a 772.14% increase in thesolution cost per control volume for 3,012.32% increase in the mesh size. For atriangular element, results displayed in Table 2a indicate that the CPU per controlvolume increases from 391�10× 10−3s to 1� 225�1× 10−3s as the grid size increases


Figure 10. Mach contours for supersonic flow over a circular arc bump at (a) Min = 1�65 and (b)Min = 1�4; and (c) comparison of predicted Mach number values along the upper and lower walls withpublished data [40] (Min = 1�4).


Figure 11. Convergence history for supersonic flow over a circular arc bump (Min = 1�4).

Figure 12. Convergence history for supersonic flow over a circular arc bump (Min = 1�65).


Figure 13. (a) Physical situation and (b) isobars for supersonic flow over an obstacle (M� = 2).


Figure 14. Convergence history for supersonic flow over an obstacle (M� = 2).

from 10,450 to 306,344 control volumes. The increase in computational cost percontrol volume in this case is 213.24% for 2,831.52% increase in mesh size.

Problem 4: Supersonic Flow over a Circular Cylinder

In this problem, depicted schematically in Figure 15a, air is approaching acircular cylinder of radius r = 1 at a supersonic Mach number of 3 (M� = 3).A uniform flow is assumed at inlet where all variables are specified. At the outlet,the flow is also supersonic and values of all variables are extrapolated from theinterior. The slip condition is used on the cylinder wall. Solutions are generatedusing quadrilateral and triangular elements over three grid networks. An illustrativegrid generated is depicted in Figure 15b. The problem has been solved by severalresearchers [40, 46–48], and the flow field is characterized by the formation of adetached shock wave at a distance d ahead of the cylinder, which can be evaluatedby the approximate relation [49].

d = 0�386 re4�67/M2� (34)

Results obtained using the coupled solver are displayed in the form of isobarsand isotherms over the domain in Figures 15c and 15d, respectively. Results arein excellent agreement with similar ones reported in [40]. The shock along thecenterline of the domain is numerically predicted to be located at d= 0�609, whichis close to the value d= 0�648 evaluated using the approximate relation (34).

The problem is solved over several grid sizes, and convergence data arepresented in Table 2b. Using quadrilateral elements, the number of iterations varies


between 95 and 284 as the grid size increases from 12,783 to 253,045 controlvolumes. For triangular elements it varies between 150 and 325 as the grid sizeincreases from 12,766 to 246,594 elements. The computational time increases forquadrilateral elements from 430 s to 28,026 s and for triangular elements from 611s to 29,027 s as the grid size increases. Moreover, the CPU per control volumefor a quadrilateral element increases from 33�64× 10−3 s to 110�76× 10−3 s whenthe grid size increases from 12,783 to 253,045 control volumes. This represents a229.25% increase in the solution cost per control volume for 1,879.54 % increase inthe mesh size. For a triangular element, the CPU per control volume increases from47�86× 10−3 s to 117�71× 10−3 s as the grid size increases from 12,766 to 246,594control volumes. The increase in computational cost in this case is 145.94% for1831.65 % increase in mesh size. The convergence history plots are displayed inFigures 16a–16c for grid networks using quadrilateral elements and in Figures 16d–16f for grid systems using triangular elements. As depicted, a smooth convergenceis obtained with quadrilateral elements, while the convergence behavior when using

Table 2. (a) Iterations and CPU time for supersonic flow over an obstacle (M� = 2), (b) Iterationsand CPU time for supersonic flow over a circular cylinder (M� = 3), (c) Iterations and CPU time forhypersonic flow over a wedge (�= 15��M� = 10), (d) Iterations and CPU time for transonic flow in aconverging-diverging nozzle [subsonic inlet at (Min = 0�3), supersonic at outlet], (e) Iterations and CPUtime for transsonic flow in a converging-diverging nozzle with a normal shock wave at x= 7 [subsonicinlet (Min = 0�3), subsonic at outlet]

Quadrilateral elements Triangular elements

Grid Size No. of #Iter. CPU�s CPU/CV Grid Size No. of #Iter. CPU�s CPU/CV

(a)10,092 180 748 74�12× 10−3 10,450 695 4� 087 391�10× 10−3

113,797 626 32� 866 288�81× 10−3 100,410 1� 737 108� 727 1082�8× 10−3

314,094 861 203� 039 646�43× 10−3 306,344 2� 071 375� 316 1225�1× 10−3

(b)12,783 95 430 33�64× 10−3 12,766 150 611 47�86× 10−3

80,159 208 8� 192 102�20× 10−3 79,218 230 8� 339 105�27× 10−3

140,601 212 11� 494 81�75× 10−3 148,054 273 14� 580 98�48× 10−3

253,045 284 28� 026 110�76× 10−3 246,594 325 29� 027 117�71× 10−3

(c)19,404 54 275 14�17× 10−3 39,470 84 953 24�14× 10−3

53,760 74 1� 091 20�29× 10−3 193,856 117 8� 157 42�08× 10−3

483,072 127 18� 148 37�57× 10−3 336,790 141 17� 849 53�00× 10−3

(d)10,000 45 86 8�60× 10−3 10,000 62 114 11�40× 10−3

100,000 140 3� 331 33�31× 10−3 100,000 168 3� 786 37�86× 10−3

300,000 217 15� 304 51�01× 10−3 300,000 270 19� 605 65�35× 10−3

(e)10,000 108 204 20�40× 10−3 10,000 185 364 36�40× 10−3

100,000 430 10� 441 104�41× 10−3 100,000 431 10� 072 100�72× 10−3

300,000 823 60� 061 200�20× 10−3 300,000 524 39� 082 130�27× 10−3


Figure 15. (a) Physical situation, (b) an illustrative grid over portion of the domain, (c) isobars, and(d) isotherms for supersonic flow over a circular cylinder (M� = 3).

triangular elements shows some oscillations even though the number of iterations isclose to that required by the corresponding cases using quadrilateral elements.

Problem 5: Hypersonic Flow over a Wedge

The physical domain is displayed schematically in Figure 17a and representsa free stream of air approaching a 15� wedge at hypersonic speed (M� = 10). Thesurface including the compression corner forms the lower boundary. The inflowboundary is placed at x = 0 and the outflow boundary is at x = 1�5. The upperboundary is chosen to be located at a height y = 1�26. The problem has been used asa benchmark by several researchers [50–53], as a one-dimensional analytical solutionto the problem exists. If the Mach number is high enough and the wedge angle issmall enough, an oblique shock wave is generated by the wedge, with the origin ofthe shock attached to the sharp leading edge of the wedge. For a given upstreamMach number M�, there is a maximum wedge angle for which the shock remainsattached to the leading edge and is given by [54, 55]

� <4

3√3 ��+ 1

(M2

� − 1)3/2

M2�(35)


Figure 16. Convergence history for supersonic flow over a circular cylinder (M� = 3).

For wedge angles greater than the maximum, a detached normal shock occurs.The upstream Mach number and wedge angle considered result in an oblique shockwave attached to the leading edge of the wedge. In this case, the analytical solutionto the problem is given by

cot �� = tan �s

[0�5 ��+ 1M2

�M2� sin2 �s− 1

− 1

](36)

M21 sin

2 �s − � = ��− 1M2� sin2 �s+ 2

2�M2� sin2 �s− ��− 1(37)

where s is the shock angle, M1 the Mach number after the shock, and � the specificheat ratio. Again the problem is solved using several grid systems of increasingdensity, and the resulting streamlines and oblique shock wave are displayed inFigure 17b. Based on Eqs. (36) and (37), the shock angle and Mach number after theshock are s = 19�94� and M1 = 5�28. The values obtained numerically are s = 19�61�

and M1 = 5�5, which are close to the analytical values given that the convective fluxis discretized with the upwind scheme and the grid is uniformly distributed over thedomain with no concentration in the region around the shock.

The convergence data are presented in Table 2c. Using quadrilateral elements,the problem is solved using three grid systems with sizes varying between 19,404and 483,072 control volumes. Results in Table 2c indicate that the number ofiterations varies between 54 and 127 and the CPU per control volume increasesfrom 14�17× 10−3 s to 37�57× 10−3 s. This represents an increase of 165.14% incomputational cost per control volume for 2,389.55% increase in grid size. For


Figure 17. (a) Physical situation and (b) streamlines for hypersonic flow over a wedge.


Figure 18. Convergence history for hypersonic flow over a wedge (M� = 10).

triangular elements, the number of iterations increases from 84 to 141 as the gridsize increases from 39,470 to 336,790 cells. The cost per control volume increases by119.55% with an increase of 753.28% in the grid size. The convergence history plotsare displayed in Figures 18a–18c for grid networks using quadrilateral elementsand in Figures 18d–18f for grid systems using triangular elements. As depicted, asmooth convergence is obtained with both quadrilateral and triangular elements,with similar convergence history plots.

Problem 6: Flow in a Converging-Diverging Nozzle

The test selected, shown schematically in Figure 19a, is a standard one thathas been used by several researchers for comparison purposes [10, 45, 56] andrepresents the flow in a converging-diverging nozzle with its cross-sectional areavarying according to

S�x = Sth + �Si − Sth(1− x

5

)2(38)

where Si = 2�035 and Sth = 1 are the inlet and throat areas, respectively, and 0 ≤x ≤ 10. Solutions are obtained for an inlet Mach number of 0.3 (Min = 0�3) for twovalues of the back pressure resulting either in a supersonic flow at exit or in theformation of a normal shock wave at x = 7. Two-dimensional numerical results forthe two cases considered are compared in terms of area-weighted pressure valueswith one-dimensional analytical solutions in Figures 19b and 19c and are shownto be in excellent agreement with numerical values, falling on top of the analyticalones.


Figure 19. (a) Physical situation and (b� c) comparison of predicted pressure values with analytical onesfor the flow in a converging-diverging nozzle with either (b) supersonic flow at exit or (c) formationof a shock wave at x = 7.


A summary of the computational data is reported in Table 2d for the casewhen the flow is supersonic in the entire diverging section of the nozzle and inTable 2e for the case when a shock wave is formed at x = 7. As shown in Table2d the number of iterations increases from 45 to 217 and from 62 to 270 asthe grid size increases from 10,000 to 300,000 control volumes with quadrilateraland triangular elements, respectively. The CPU per cell increases by 493.14% and473.25% as the grid increases by 2,900% for quadrilateral and triangular elements,respectively. When a shock develops in the domain Table 2e, the number ofiterations varies between 108 and 823 for quadrilateral elements and between 185and 524 for triangular elements. The cost per control volume increases by 881.37%for quadrilateral elements and 257.88% for triangular elements as the grid sizeincreases by 2,900%.

The convergence history plots are displayed in Figures 20 and 21 for the casewhen the flow at the exit from the nozzle is supersonic and when a shock wave ispresent, respectively. For fully supersonic flow in the diverging section, convergencehistories are displayed in Figures 20a–20c for grid networks using quadrilateralelements and in Figures 20d–20f for grid systems using triangular elements. Asdepicted, curves are very similar and indicate that convergence starts slow untila good initial guess is obtained, beyond which residuals drop very quickly. Forthe case when a shock occurs in the diverging section Figures 21a–21f , a cyclicreduction in the residuals is noticed while the solver is trying to home on the exactlocation of the shock. For a given grid size, the convergence behavior is seen to bealmost independent of the type of element used.

Figure 20. Convergence history for transsonic flow in a converging-diverging nozzle [subsonic inlet(Min = 0�3], supersonic at outlet).


Figure 21. Convergence history for transsonic flow in a converging-diverging nozzle with a normalshock wave at x = 7 (Min = 0�3).

CLOSING REMARKS

A newly developed, fully coupled, pressure-based algorithm for the solutionof incompressible and compressible flow problems was presented. The performanceof the solver was assessed by presenting solutions to several flow problems withspeeds varying from low subsonic to hypersonic values. For each problem, solutionswere generated over several grid systems ranging from 10,000 to almost 400,000control volumes. Results presented in the form of convergence history plots,required number of iterations, total CPU time, and CPU time per control volumedemonstrated the robustness and high stability of the solver.

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